Review of Statistical Estimation

Introduction

• In this lecture, we will review summary statistics

  • Continuous variables

    • Mean
    • Median
    • Percentiles / quartiles
    • Variance and standard deviation
    • Interquartile range

  • Categorical variables

    • Count
    • Percentage

Summary Statistics: Introduction

• In this course, we will review formulas, but we will use R for computational purposes

  • Remember to refer to the lecture notes for specific code needed

  • Code is also available on this course’s GitHub repository

• We can use base R for some things, but I try to stay in the tidyverse when possible.

• If we need to install packages, we use the install.packages() function,

install.packages("package name")

• To call packages in, we use the library() function,

library(tidyverse)

Summary Statistics: Mean

• Definition: sample arithmetic mean

\bar{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}

• R syntax:

# method 1: base R
mean(vector_name) # if your variable is not part of a dataset
mean(dataset_name$variable_name) # if your variable is part of a dataset

# method 2: tidyverse
dataset_name %>% summarize(mean(variable_name))

Summary Statistics: Mean

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the average exam score.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
mean(scores)

[1] 79

Summary Statistics: Mean

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the average exam score.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
mean(scores)

[1] 79

• What happens to the mean if there is an extreme observation?

scores <- c(82, 77, 90, 71, 26, 68, 74, 84, 94, 88)
mean(scores)

[1] 75.4

Summary Statistics: Variance and Standard Deviation

• Definition: sample variance

s^2 = \frac{\sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n}}{n-1}

• Definition: sample standard deviation

s = \sqrt{s^2}

• R syntax:

# method 1: base R
var(vector_name) or sd(vector_name) # if your variable is not part of a dataset
var(dataset_name$variable_name) or sd(dataset_name$variable_name) # if your variable is part of a dataset

# method 2: tidyverse
dataset_name %>% summarize(var(variable_name), sd(variable_name))

Summary Statistics: Variance and Standard Deviation

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the variance and standard deviation.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
var(scores)

[1] 107.1111

sd(scores)

[1] 10.34945

Summary Statistics: Variance and Standard Deviation

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the standard deviation.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
sd(scores)

[1] 10.34945

• What happens to the standard deviation if there is an extreme observation?

scores <- c(82, 77, 90, 71, 26, 68, 74, 84, 94, 88)
sd(scores)

[1] 19.30573

Summary Statistics: Median

• Definition: median

  • The value that lies in the middle of the data when arranged in ascending order.

    • If n is odd, then the median is literally the middle number.

    • If n is even, then the median is the average of the two middle numbers.

• R syntax:

# method 1: base R
median(vector_name) # if your variable is not part of a dataset
median(dataset_name$variable_name) # if your variable is part of a dataset

# method 2: tidyverse
dataset_name %>% summarize(median(variable_name))

Summary Statistics: Median

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the median exam score.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
median(scores)

[1] 79.5

Summary Statistics: Median

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the median exam score.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
median(scores)

[1] 79.5

• What happens to the median if there is an extreme observation?

scores <- c(82, 77, 90, 71, 26, 68, 74, 84, 94, 88)
median(scores)

[1] 79.5

Summary Statistics: Percentiles

• Definition: k^th percentile, P_k

  • k% of the observations in the dataset are less than or equal to that value.

• R syntax:

quantile(vector_name, percentile)  # if your variable is not part of a dataset
quantile(dataset_name$variable_name, percentile) # if your variable is part of a dataset

Summary Statistics: Percentiles

• The following data represent the first exam score of 10 randomly selected students in STA2023.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)

• What is the 45th percentile?

quantile(scores, 0.45)

  45% 
77.25 

• What is the 90th percentile?

quantile(scores, 0.90)

 90% 
90.4 

Summary Statistics: Quartiles

• Definition: quartiles

  • Values that divide the dataset into fourths, or four equal parts: P₂₅, P₅₀ (median), P₇₅

• Definition: five number summary

  • Minimum, P₂₅, P₅₀ (median), P₇₅, maximum

• R Syntax:

quantile(vector_name, c(0.00, 0.25, 0.50, 0.75, 1.00))  # if your variable is not part of a dataset
quantile(dataset_name$variable_name, c(0.00, 0.25, 0.50, 0.75, 1.00)) # if your variable is part of a dataset

Summary Statistics: Quartiles

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the five number summary.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
quantile(scores, c(0.00, 0.25, 0.50, 0.75, 1))

   0%   25%   50%   75%  100% 
62.00 71.75 79.50 87.00 94.00 

Summary Statistics: Quartiles

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the five number summary.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
quantile(scores, c(0.00, 0.25, 0.50, 0.75, 1))

   0%   25%   50%   75%  100% 
62.00 71.75 79.50 87.00 94.00 

• What happens to the five number summary if there is an extreme observation?

scores <- c(82, 77, 90, 71, 26, 68, 74, 84, 94, 88)
quantile(scores, c(0.00, 0.25, 0.50, 0.75, 1))

   0%   25%   50%   75%  100% 
26.00 71.75 79.50 87.00 94.00 

Summary Statistics: Interquartile Range

• Definition: interquartile range

  • A measure of the spread of the middle half of the data

• R syntax:

# method 1: base R
IQR(vector_name) # if your variable is not part of a dataset
IQR(dataset_name$variable_name) # if your variable is part of a dataset

# method 2: tidyverse
dataset_name %>% summarize(IQR(variable_name))

Summary Statistics: Interquartile Range

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the IQR.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
IQR(scores)

[1] 15.25

Summary Statistics: Interquartile Range

• The following data represent the first exam score of 10 randomly selected students in STA2023. Find the IQR.

scores <- c(82, 77, 90, 71, 62, 68, 74, 84, 94, 88)
IQR(scores)

[1] 15.25

• What happens to the IQR if there is an extreme observation?

scores <- c(82, 77, 90, 71, 26, 68, 74, 84, 94, 88)
IQR(scores)

[1] 15.25

Summary Statistics: Frequency Tables

• When we are dealing with categorical data, we summarize using frequency tables.

  • We are interested in the count and percentage

• e.g., from the UWF Fact Book, in Fall 2021, there were

  • 1273 (14.4%) freshmen
  • 1349 (15.2%) sophomores
  • 2431 (27.4%) juniors
  • 3807 (43.0%) seniors

• R syntax:

dataset_name %>%
  group_by(var1_name, var2_name) %>% # do not change anything under this
  summarise(n = n()) %>%
  mutate(freq = n / sum(n))

Summary Statistics: Frequency Tables

• Consider the Motor Trends car road tests data, built into R.

• The data was extracted from the 1974 Motor Trend magazine, and includes aspects of car design and performance for 32 cars (1973-74 models).

data("mtcars")
head(mtcars, n=5)

                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2

Summary Statistics: Frequency Table

• Let’s find the frequency tables for the number of forward gears (gear) and the type of transmission (am; automatic vs. manual),

mtcars %>%
  group_by(am, gear) %>%
  summarise(n = n()) %>%
  mutate(freq = n / sum(n))

# A tibble: 4 × 4
# Groups:   am [2]
     am  gear     n  freq
  <dbl> <dbl> <int> <dbl>
1     0     3    15 0.789
2     0     4     4 0.211
3     1     4     8 0.615
4     1     5     5 0.385

• Note that these are overall frequencies.

Data Visualization: Introduction

• When presenting results to others, sometimes it is helpful to create a visualization.

• Continuous data:

  • Histogram (one variable)
  • Scatterplot (two variables)

• Categorical data:

  • Bar charts

• Related to analyses:

  • Confidence intervals
  • Regression lines

• We can also use color to incorporate other variables

  • e.g., scatterplot with weight on y-axis, age on x-axis, markers colored by diabetic status

Data Visualization: Introduction

• We will use the ggplot2 package for most of our graphing needs.

  • This package is loaded in with the tidyverse package.

• A good reference book is the official ggplot2: elegant graphics for data analysis text.

  • You can buy a hard copy, however, it is available online for free.

• I will often google keywords + ggplot2 and look for examples that provide code.

  • e.g., “histogram ggplot2” led me to this website

  • e.g., “change color of dot ggplot2” led me to this website

  • Sometimes I have to look at several links before I find what I am looking for.

Data Visualization: Introduction

• We start with the ggplot() function to specify our underlying canvas.

ggplot()

Data Visualization: Introduction

• We will use the tidyverse pipe operator (%>%) to pipe data into the ggplot() function.

dataset %>% ggplot()

• Then, we will specify the aesthetics using aes() inside of ggplot().

mtcars %>% ggplot(aes(x = hp, y = mpg))

Data Visualization: Introduction

• We will add elements to our graph using geom_ functions.

  • geom_line() creates a line
  • geom_point() creates a scatterplot
  • geom_bar() creates a bar chart
  • geom_text() puts text on the graph
  • You can find a list of geom_ functions on the tidyverse website

• The order that you add them matters!

  • geom_line() + geom_point() = points on top of line
  • geom_point() + geom_line() = line on top of points

Data Visualization: Introduction

• We can also customize every aspect of our graphs.

  • e.g., the default background is gray, but I personally do not like it, so I typically use theme_minimal() or theme_bw() to give a white background

  • e.g., we can increase the font size to make things readable

  • e.g., we can specify colors for: markers (dots/points), outline of a bar chart or histogram, filling of a bar chart or histogram, lines, text, etc.

• There are additional functions within other (non-tidyverse) packages that will help us with customization.

  • We can put graphs together using the ggarrange() function in the ggpubr package

  • We can use geom_emoji() from the the emoGG to display emojis in graphs :)

Data Visualization: Introduction

• I do not expect you to become an expert in data visualization

• As with other R code, I will provide basic code during lecture

• I do encourage curiosity and exploring further

• R is a very, very powerful tool for graphing!

  • Even before I was An Official R Programmer^©, I used ggplot2 to construct graphs.

  • Other programs are just not great. :(

• Today we will look at graphs that go along with summary statistics, but we will learn other ways to graph data as we progress through the semester.

Data Visualization: Histogram

ex1 %>% 
  ggplot(aes(x=value)) + 
  geom_histogram(bins=50) +
  labs(x = "value of variable",
       y = "number of observations") +
  theme_bw() 

Data Visualization: Histogram

ex2 %>% 
  ggplot(aes(x=value)) + 
  geom_histogram(bins=50) +
  labs(x = "value of variable",
       y = "number of observations") +
  theme_bw() 

Data Visualization: Histogram

ex3 %>% 
  ggplot(aes(x=value)) + 
  geom_histogram(bins=50) +
  labs(x = "value of variable",
       y = "number of observations") +
  theme_bw() 

Data Visualization: Histogram

ex4 %>% 
  ggplot(aes(x=value)) + 
  geom_histogram(binwidth = 0.1) +
  labs(x = "value of variable",
       y = "number of observations") +
  theme_bw() 

Data Visualization: Histogram

ex4 %>% 
  ggplot(aes(x=value)) + 
  geom_histogram(binwidth = 0.1, color = "#003865", fill = "#8DC8E8") +
  labs(x = "value of variable",
       y = "number of observations") +
  theme_bw() 

Data Visualization: Scatterplot

mtcars %>% 
  ggplot(aes(y = mpg, x = hp)) + 
  geom_point() + 
  labs(x = "Horsepower",
       y = "Gas Mileage") +
  theme_bw() 

Data Visualization: Scatterplot

mtcars %>% 
  ggplot(aes(y = mpg, x = hp)) + 
  geom_point(size = 5) + 
  labs(x = "Horsepower",
       y = "Gas Mileage") +
  theme_bw() 

Data Visualization: Scatterplot

mtcars %>% 
  ggplot(aes(y = mpg, x = hp, color = am)) + 
  geom_point(size = 5) + 
  labs(x = "Horsepower",
       y = "Gas Mileage") +
  theme_bw() 

Data Visualization: Scatterplot

mtcars %>% 
  ggplot(aes(y = mpg, x = hp, color = as.factor(am))) + 
  geom_point(size = 5) + 
  labs(x = "Horsepower",
       y = "Gas Mileage") +
  theme_bw() 

Data Visualization: Scatterplot

mtcars %>% 
  ggplot(aes(y = mpg, x = hp, color = as.factor(am))) + 
  geom_point(size = 5) + 
  labs(x = "Horsepower",
       y = "Gas Mileage",
       color = "Transmission") +
  scale_color_manual(labels = c("Automatic", "Manual"),
                     values = c("#003865", "#8DC8E8")) +
  theme_bw() 

Data Visualization: Plot of Means

means <- mtcars %>%
  group_by(cyl) %>%
  summarize(mean = mean(mpg)) %>%
  ungroup()

means %>% 
  ggplot(aes(y = mean, x = cyl)) + 
  geom_point(size = 5) + 
  labs(x = "Number of Cylinders",
       y = "Average Gas Mileage") +
  theme_bw() 

Data Visualization: Plot of Means

Data Visualization: Plot of Means

means <- mtcars %>%
  group_by(cyl, am) %>%
  summarize(mean = mean(mpg)) %>%
  ungroup()

means %>% 
  ggplot(aes(y = mean, x = cyl, color = as.factor(am))) + 
  geom_point(size = 5) + 
  labs(x = "Horsepower",
       y = "Gas Mileage",
       color = "Transmission") +
  scale_color_manual(labels = c("Automatic", "Manual"),
                     values = c("#003865", "#8DC8E8")) +
  theme_bw() 

Data Visualization: Plot of Means

Data Visualization: Plot of Means

means <- mtcars %>%
  group_by(cyl, am) %>%
  summarize(mean = mean(mpg),
            sd = sd(mpg)) %>%
  ungroup()

means %>% 
  ggplot(aes(y = mean, x = cyl, color = as.factor(am))) + 
  geom_point(size = 5) + 
  geom_errorbar(aes(ymin=mean-sd, ymax=mean+sd), width = 0.15) +
  labs(x = "Horsepower",
       y = "Gas Mileage",
       color = "Transmission") +
  scale_color_manual(labels = c("Automatic", "Manual"),
                     values = c("#003865", "#8DC8E8")) +
  theme_bw() 

Data Visualization: Plot of Means

Data Visualization: Bar Chart

mtcars %>% ggplot(aes(x=as.factor(cyl))) +
  geom_bar() +
  labs(x = "Number of Cylinders",
       y = "Number of Cars") +
  theme_bw()

Data Visualization: Bar Chart

mtcars %>% ggplot(aes(x=as.factor(cyl), fill=as.factor(am))) +
  geom_bar(position = 'dodge') +
  labs(x = "Number of Cylinders",
       y = "Number of Cars",
       fill = "Transmission") +
  scale_fill_manual(labels = c("Automatic", "Manual"),
                     values = c("#003865", "#8DC8E8")) +
  theme_bw()

Wrap Up

• In lecture, we have reviewed how to describe data.

  • Summary statistics
  • Basic data visualization

• There is not a one-size-fits-all graph!

  • Always keep in mind what is the story we are trying to tell and what aids in our explanation.

• Next, we will review statistical inference.

  • Confidence intervals
  • Hypothsis testing